Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1895 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,15)(16,21,24,19)(17,20,25,22)(23,27,28,26)(29,30,34,32)(31,37,38,35)(33,40,41,36)(39,42,44,43);(5,6,9,7)(8,10,12,11)(14,15)(17,25)(19,21)(23,28)(31,38)(33,41)(35,37)(36,40);(6,7)(8,12)(13,15,18,14)(16,21,24,19)(17,20,25,22)(23,26,28,27)(29,31)(30,35)(32,37)(33,39)(34,38)(36,42)(40,43)(41,44);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27)(29,32,34,30)(31,35,38,37)(33,40,41,36)(39,42,44,43);(5,8,9,12)(6,10,7,11)(13,17)(14,20)(15,22)(16,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,41)(30,36,32,40)(31,39,38,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=gh,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5385150);